Randić energy of digraphs.

We assume that is a directed graph with vertex set and arc set . A VDB topological index of is defined as where and denote the outdegree and indegree of vertices and , respectively, and is a bivariate symmetric function defined on nonnegative real numbers. Let be the general adjacency matrix defined as if , and 0 otherwise. The energy of with respect to a VDB index is defined as , where are the singular values of the matrix . We will show that in case is the Randić index, the spectral norm of is equal to 1, and rank of is equal to rank of the adjacency matrix of . Immediately after, we illustrate by means of examples, that these properties do not hold for most well-known VDB topological indices. Taking advantage of nice properties the Randić matrix has, we derive new upper and lower bounds for the Randić energy in digraphs. Some of these generalize known results for the Randić energy of graphs. Also, we deduce a new upper bound for the Randić energy of graphs in terms of rank, concretely, we show that for all graphs , and equality holds if and only if is a disjoint union of complete bipartite graphs.